Graphing in Intercept Form
The intercept form of a quadratic function is \(f\left(x\right)=a\left(x-p\right)\left(x-q\right)\). Explore the graph of the function in intercept form to the parent function \(y = x^2\) and answer the following questions.
The intercept form of a quadratic function is \(f\left(x\right)=a\left(x-p\right)\left(x-q\right)\). Explore the graph of the function in intercept form to the parent function \(y = x^2\) and answer the following questions.
Quick Check
1. How does the \(a\) value affect the graph when compared to the parent function? a. If \(a > 1\)? b. If \(0 < a < 1\)? c. If \(a < 0\) 2. How are \(p\) and \(q\) related to the graph? Quick Check Solutions |
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To accurately graph a quadratic function in intercept form, you probably want to plot at least five points, including the two x-intercepts and the vertex.
- The x-coordinate of the vertex is midway between the two x-intercepts and is found using the formula \(x=\large\frac{\left(p+q\right)}{2}\).
- The y-coordinate of the vertex is found by evaluating the function at the x-coordinate of the vertex.
We can say the vertex of the quadratic function in intercept form is \(\large\left(\frac{\left(p+q\right)}{2},f\left(\frac{\left(p+q\right)}{2}\right)\right)\).
To graph a quadratic function in intercept form:
- Find and plot the two x-intercepts. These are found by setting each factor equal to \(0\) and solving for \(x\).
- Find the vertex and plot it.
- Draw in the axis of symmetry.
- Evaluate the function at a value of \(x\) to find another point.
- Reflect the point over the axis of symmetry.
Keep in mind, if \(a > 0\) (positive) the parabola opens up and if \(a < 0\) (negative) the parabola opens down.
Let’s look at a couple of examples.
Example 1: Graph the quadratic function \(f\left(x\right)=-2\left(x-4\right)\left(x+6\right)\). Identify the vertex, axis of symmetry, domain, range, and maximum or minimum value. (Some of the work is shown below, but watch the video to see the full explanation and actual graph).
Find the x-intercepts
The factors are \(\left(x-4\right)\) and \(\left(x+6\right)\).
\(x-4=0\) and \(x+6=0\)
\(x=4\) and \(x=-6\)
The x-intercepts are \(\left(4,0\right)\) and \(\left(-6,0\right)\).
Find the vertex and the axis of symmetry
\(x=\frac{\left(4+-6\right)}{2}\)
\(x=\frac{-2}{2}\)
\(x=-1\)
\(f\left(-1\right)=-2\left(-1-4\right)\left(-1+6\right)\)
\(f\left(-1\right)=-2\left(-5\right)\left(5\right)\)
\(f\left(-1\right)=50\)
The vertex is \(\left(-1,50\right)\) and the axis of symmetry is \(x=-1\).
Find another point on the graph
Let \(x = 0\)
\(f\left(0\right)=-2\left(0-4\right)\left(0+6\right)\)
\(f\left(0\right)=-2\left(-4\right)\left(6\right)\)
\(f\left(0\right)=48\)
The coordinate \(\left(0,48\right)\) is a point on the graph.
Plot the intercepts, the vertex and the point \(\left(0,48\right)\) and reflect this last point over the axis of symmetry.
The vertex is \(\left(-1,50\right)\).
The axis of symmetry is \(x=-1\).
The domain is \(\left\{x\mid x\in R\right\}\) which can also be represented as \(\left(-\infty,\infty\right)\).
The range is \(\left\{y\mid y\le50\right\}\) which can also be represented as \(\left(-\infty,50\right]\).
There is a maximum value at \(y=50\) when \(x=-1\).
Example 2: Graph the quadratic function \(f\left(x\right)=\frac{1}{2}\left(x+2\right)\left(x-4\right)\). Identify the vertex, axis of symmetry, domain, range, and maximum or minimum value.